How to get Blender to add new face-ids but not recalculate old face-ids?
Geometry element indices can change at any time, don't rely on them!

See e.g. How are vertex indices determined?

You should instead use a custom data layer to store the group (continent) information in. I would suggest bm.faces.layers.int. How to display them in a panel is a bit tricky, because you need to iterate the entire mesh before you know which numbers / continents exist.

API docs: http://www.blender.org/api/blender_python_api_2_75_release/bmesh.html#customdata-access

Note that if a user extrudes a face that is part of a continent already, then that property might be copied to the new faces. It's definitely true for vertex extrusion, but I'm not entirely sure when or which faces may receive the custom properties of their ancestors. Also see second half of my answer here: https://blender.stackexchange.com/a/36816/1363

Another approach would be to use Vertex groups. There are some downsides however: Vertices can be assigned with weights, they can be assigned to multiple groups and the face selection may differ for certain topologies (3 adjacent faces, left and right part of a continent, the middle one not will select all 3 if vertex groups are used, since all of the vertices are used by the left and right face).